Local Definability of $\mathsf{HOD}$ in $L(\mathbb{R})$

TL;DR

This paper demonstrates that within the model $L(R)$, under large cardinal assumptions, the structure of $ ext{HOD}$ is locally definable from its initial segments at all $ ext{HOD}$-cardinals in a specific range, elaborating on Steel's core model statement.

Contribution

It establishes the local definability of $ ext{HOD}$ in $L(R)$ from initial segments at all relevant $ ext{HOD}$-cardinals, extending previous core model results.

Findings

$ ext{HOD}$ is locally definable from initial segments at all $ ext{HOD}$-cardinals in the specified range.

The result holds assuming large cardinal axioms within $L(R)$.

Provides a detailed elaboration of Steel's statement on $ ext{HOD}$ as a core model.

Abstract

We show that in $L(\mathbb{R})$, assuming large cardinals, $\mathsf{HOD} {\parallel}\eta^{+\mathsf{HOD}}$ is locally definable from $\mathsf{HOD} {\parallel}\eta$ for all $\mathsf{HOD}$-cardinals $\eta\in [\boldsymbol{\delta}^2_1,\Theta)$. This is a further elaboration of the statement "$\mathsf{HOD}^{L(\mathbb{R})}$ is a core model below $\Theta$" made by John Steel.